A Multi-Class Multi-Mode Variable Demand Equilibrium Model
A Multi-Class Multi-Mode Variable Demand Network Equilibrium Model
With Hierarchical Logit Structures
Michael Florian1,2, Jia Hao Wu1,2 , Shuguang He1
INRO Solutions Inc1
Center for Research on Transportation, Université de Montréal 2
April 1999
Abstract. We consider a multi-class multi-mode variable demand network equilibrium model where the mode
choice model is given by aggregate hierarchical logit structures and the destination choice is specified as a multiproportional entropy type trip distribution model. The travel time of transit vehicles depends on the travel time of other
vehicles using the road network. A variational inequality formulation captures all the model components in an
integrated form. A solution algorithm, based on a Block Gauss-Seidel decomposition approach coupled with the
method of successive averages results in an efficient algorithm which successively solves network equilibrium models
with fixed demands and multi-dimensional trip distribution models. Numerical results obtained with an implementation
of the model with the EMME/2 software package are presented based on data originating from the city of Santiago,
Chile.
Key words: network equilibrium, variable demand, hierarchical logit structure, simultaneous equilibrium.
1. Introduction
Over the past twenty years, the models used in transportation planning for congested urban networks received a lot of
attention from researchers and practitioners. One of the principal issues studied and debated is that of the consistency of
the models used for predicting the demand for travel and the network models used to determine the levels of service.
One seeks to ensure that the levels of service which are used to compute the demand would be reproduced if that
demand were used again to compute the levels of service. The keywords that we find in the literature regarding these
models are "integrated", "combined", "simultaneous", "feedback" and "variable demand". The recent San Francisco
Bay Area lawsuit (see Gattett and Wachs, 1996, p. 199) shows that the importance of this issue has social and political
impacts.
The model presented in this paper is inspired from the work carried out for the ESTRAUS project in Santiago, Chile
(ESTRAUS, 1985-1998) and is a complex multi-class multi-mode variable demand network equilibrium model with
hierarchical structures which, until now, was not formulated in an integrated way as a variational inequality
formulation. This formulation permits the rigorous mathematical analysis of the model structure and the development
of an efficient solution algorithm.
The remainder of the paper is organized as follows. In the next section, we provide a literature review of the related
problems. In section 3, we introduce the notations and develop the variational inequality formulation. Section 4
provides the analysis of the mathematical structure of the model by using the corresponding KKT conditions, and the
solution algorithm is given in section 5. In section 6, we provide computational results, and some conclusions are
presented in the last section.
2. A Brief Literature Review
The development of network equilibrium models that may be formulated as optimization problems originates with the
seminal contribution of Beckmann, Winsten and McGuire (1956). The introduction of variable demand network
equilibrium models for one mode, where the destination choice is determined by entropy type distribution models, can
be found in the contributions of Florian, Ferland and Nguyen (1975), and Evans (1976).
*
This project is supported in part by Chile University.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
This model, referred to as combined trip distribution and assignment models, was extended to two modes and mode
choice by Florian and Nguyen (1978) when the travel times by two modes, say auto and transit, are not related. These
models are convex cost multi-commodity network optimization models that may be solved by the linear approximation
method (Franke and Wolfe, 1956) or by the partial linear approximation method first suggested by Evans (1976).
Florian (1977) formulated a two-mode network equilibrium model where the transit travel times depend on the auto
travel times, and the auto travel times account for the pressure of the transit vehicles which share the capacity of the
roads.
This model was recognized to be a special instance of a variational inequality formulation following the work of Smith
(1979) and Dafermos (1980) for network equilibrium models with asymmetric link cost functions. It was further
analyzed by Fisk and Nguyen (1982) and Florian and Spiess (1982) and solved by adaptation of the nonlinear Jacobi
method. Sufficient conditions for the convergence of the algorithm were developed by Dafermos (1982) and Pang and
Chan (1982). Friesz (1980) developed an optimization formulation of the multiclass combined trip distribution, mode
choice and assignment model where the travel costs depend on the mode chosen, and Fisk and Boyce (1983) developed
a variational inequality formulation for this model where the travel costs are not mode interrelated and the mode choice
is given by a multinomial logit function. In the attempt to introduce the trip generation model in an integrated model,
Safwat and Magnanti (1988) developed an integrated model which has an equivalent convex cost optimization
formulation, and Safwat, Nobil and Walton (1988) provide a calibration and computational experience with this model.
Lam and Huang (1992) present a multiclass combined trip distribution mode choice on assignment model formulated as
well as an equivalent convex cost optimization problem and solve it with the partial linearization method. The first
formulation of a network equilibrium model with hierarchical logit demand functions is that of Fernandez et al. (1994)
where the choice of stations for the "park-n-ride" mode is given by a lower nest of the demand function. The model is
formulated as an equivalent convex cost optimization model and the solution method suggested is an adaptation of the
partial linear approximation algorithm. Oppenheim (1995) presents several versions of combined trip distribution mode
choice and assignment models. The importance of properly stating and solving integrated network equilibrium models
is stressed by Boyce (1998). A recent contribution is that of Abrahamsson and Lundqvist (1999) who developed a
convex cost optimization model for the combined trip distribution, mode choice and assignment models with
hierarchical choices, where distribution may precede mode choice or vice versa. In their model, the transit travel times
are independent of the auto travel times.
The model developed in this paper is motivated by the ESTRAUS model developed for the city of Santiago, Chile and
described in various internal technical reports. In this model, the trip productions are given by class and purpose of
travel. The mode choice model is an aggregated hierarchical logit function (Ortuzar and Willumsen (1996, p. 219).
There are thirteen classes of travelers, three travel purposes and multiple modes of travel, which include walk, auto,
multiple transit modes and combined modes ("park-n-ride"). The model components were calibrated by using survey
data, and a computational procedure was used for its numerical solution without the statement of an integrated model.
In this paper, we formulate the model as a variational inequality formulation and show that the Karush-Kuhn-Tucker
(KKT) conditions replicate all the model components. Then a solution algorithm based on a Block Gauss-Seidel
decomposition of the model which is akin to the partial linear approximation method of Evans (1976) is used to
develop a solution algorithm.
3. Model Formulation
In order to present the model components and the variational inequality formulation, the following notation is used.
The arcs of the road network are designated by a , a  A , where A is the set of arcs. The segments of the transit lines
which compare the transit networks are designated b, b  B , where B is the set of segments of the transit network.
a ( b ) designated that segment b of a transit line segment uses link a of the road network. The demand for travel by
user class m, m  M , for purpose p, p  P , class n, n  N and mode m, m  M for the origin-destination pair (i, j ) , is
denoted Tijpnm where P, N , M are the sets of purposes, classes and modes. This demand may use paths Rijpnm  R where
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
R is the set of all routes
R
R
pnm
ij
( i , j ), p ,n ,m
. It is also useful to identify M1 as the subset of modes which use private
vehicles (auto, taxi), M 2 as the subset of transit related modes and M 3 as the walk mode.
A hierarchical logit structure for each purpose has three levels: the root, the groups and the modes. G p denotes a set of
groups or nests of nodes for purpose p , and g denotes a set of modes in a group or nest. These are illustrated in
l
l q
q
l q
l
Figure 1, where G p  g1 , g2 , g3 , g1  m1 , g2  m2 and g3  m3 , m4 , m5
q
Gp
Groups
g3
g2
g1
Modes
m1
m2
m3
m4
m5
Figure 1. Nested logit structure for purpose p .
The inclusion coefficients of each group are given as  gp 
by origin i , purpose p and class n as
R
S
T
1 if g 1
 gp if otherwise
all g, p . The production of trips are specified
Oipn while the attractions of trips are specified by destination j and purpose
p , D jp . The travel demands Tijpnm give rise to path flows hrpnm , r  Rijpnm . The feasible  region of the demands for
travel and the path flows is stated next, where dual variables, which will be used later in the analysis are indicated in
brackets.
 T
 Oipn , i, p, n
png
ij
g
( ipn )
(1)
( ip )
(2)
( Lijpng )
(3)
(  ijpnm )
(4)
( rpnm )
(5)
j
 T
png
ij
g
n
 D jp , j, p
i
T
 Tijpng  0, j, p, n, g
pnm
ij
mg
 gp (
h
pnm
r
r R m
pnm
hr  0,
Tijpnm  0,
Tijpng
 Tijpnm )  0, j, p, n, m  g
r, p, n, m
( i, j ), p, n, m
(6)
 0, ( i, j ), p, n, g
(7)
Equations (1) and (2) are the balance equations for productions and attractions. Equation (3) relates mode totals to
group totals. Equation (4) is the correction of flow equations for path flows. Note that the multiplication by  gp is
required for the analysis to follow. Equations (5), (6) and (7) are the usual positive and nonegativity requirements on
path flows and demands respectively.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
The presentation method that we have chosen is to state the variational inequality formulation first and then to show
that the KKT conditions recover the model components. The variational inequality formulation supposes that  pn is a
calibrated parameter for all (p,n) and  gp  ( 0 ,1) is the inclusion coefficient.
Find (h*,T*)   which satisfies
  [  (  ( 
pn
( ij )
g G p
mg
p pnm *
( h , T * )( hr
g Cr
 hr* )) 

p
g
*
ln(Tijpnm* / Tijpng* )(Tijpnm  Tijpnm )) 
mg
r
1
 pn
ln Tijpng* (Tijpng  Tijpng* )]  0,
( h, T ) .
(8)
It is worthwhile to note that (8) may be rewritten in the equivalent form:
  [  ( 
pn
( ij )
g G
p
mg
p
g
Crpnm ( h* , T * )( hr  hr* ))    gp ln(Tijpnm* )(Tijpnm  Tijpnm ))  (
*
mg
r
1
 pn
  gp )ln Tijpng* (Tijpng  Tijpng* )]  0,
( h, T )  .
(9)
This formulation looks similar to some mathematical structures of combined trip distribution mode choice and
assignment models which have equivalent convex cost optimization formulations. See Fernandez et al. (1994) and
Boyce (1998, pp. 81). Although the model presented here does not have an equivalent convex cost optimization
formulation, the variational inequality approach provides a powerful way to analyze the model structure.
4. The analysis of the mathematical structure of the model
In this section, we derive and analyze the KKT conditions of the variational inequality formulation (35), which are
given below .
Tijpng :
1
 pn
ln Tijpng   ipn   ip  Lijpng  0
(10)
Tijpnm :
 gp ln(Tijpnm / Tijpng )  Lijpng   gp  ijpnm  0
(11)
hrpnm:
 gp Crpnm   gp  ijpnm   rpnm  0
(12)
Complementarity: hrpnm rpnm  0

pnm
r
(13)
 0 and (1)-(7).
From (10), we have
Tijpng  e
 pn ( ipn  pj )   pn Lijpng
e
.
(14)
From (11), we have
Tijpnm / Tijpng  e
(1/  gp ) Lijpng   ijpnm
e
, m g .
(15)
Using (15) and (3), we have
e
(1/  gp ) Lijpng
  e
  ijpnm'
(16)
m'g
which implies
Lijpng   gp ln(  e
m'g
  ijpnm'
(17)
)
From (14) and (15), it follows that
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
Tijpnm  e
 pn ( ipn  pj ) (1/  gp   pn ) Lijpng   ijpnm
e
e
.
(18)
The analysis of (10) - (18) leads to the following propositions which emphasize the fact that the variational inequality
png
formulation (8) implies both group and mode proportions ( Pij
, Pijpnm ) of demands computed by the hierarchical logit
structure as well as the network equilibrium path choice conditions.
Proposition 1(Group and Mode Proportions)
Tijpng

Pijpng
Tijpn

e
  pn Lijpng
e

  pn Lijpng '
e
  pn Lijpng
e
  pn Lijpn
,
g G p
(19)
g ' G p
and
Tijpnm

Pijpnm

Tijpng
e
 uijpnm
e
 uijpnm'
e

e
m'g
 uijpnm
 (1/  gp ) Lijpng
, m  M.
(20)
Proposition 2 (Multinomial Logit Model)
If  gp  1( g  G p ), then we obtain the traditional multinomial logit structure for purpose p, i.e.,
Tijpnm
Pijpnm 

Tijpn
e
 uijpnm

e
 uijpnm'
, m  M.
m'M
Proposition 3 (Multi-Class Multi-Mode Equilibrium Conditions)
Crpnm
R
|S 
|T 
pnm
ij
pnm
ij
if
hrpnm  0
if
hrpnm  0
, r , p, n, m  M
□
Proof: Conditions (12) and (13) imply the following equilibrium conditions.
The KKT conditions are also used for the algorithm development. It is important to recall the way that costs are
propagated upward in the hierarchical logit demand structure. We define Lijpng   gp ln(  e
m'g
  ijpnm'
) , (i, j ), p, n, g
as group logsum for purpose p, class n and group g (this definition is particular to the ESTRAUS model which
inspired this model), where  gp is the inclusion coefficient of group g (see Ortuzar and Willumsen, 1996, pp. 219).
We define Lijpn  
1
 pn
ln 
e
  pn Lijpng
, (i, j ), p, n as top logsum for purpose p and class n. The total demands for
g G p
each p, n may be obtained as the solution of the following multi-proportional matrix balancing problem (Combined
trip distribution/mode choice model), by noting that Tijpn 
T
png
ij
g G
Tijpn  e
T
pn
ij
,
p
 pn ( ipn  pj )   pn Lijpn
e
(21)
 Oipn , i, p, n
(22)
j
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
 T
pn
ij
n
 D jp , j, p
(23)
i
Tijpn  0, ij, p, n
(24)
which may be easily shown to be equivalent to the following entropy maximization problem:
max 

pn
s.t.
T
pn
ij
w
1
 pn
Twpn (ln Twpn  1)
 Oipn , i, p, n
j
 T
pn
ij
n
 D jp , j, p
i
 T
pn pn
ij Lij
i
 L pn , p, n
(25)
j
Tijpn  0, ( i, j ), p, n
where L pn is a constant (unknown) such as total generalized cost, and the dual variable associated with (25) is set to be
1. (This is exactly the trip distribution component of the ESTRAUS model.)
5. A Solution Algorithm
In this section, we develop the solution algorithm used to obtain the solution of the variational inequality (9). We
employ a Block Gauss-Seidel method (Equilibration Algorithm) where the network equilibrium formulation for given
demands are computed in one block and the trip distribution/mode choice computations are computed in the other. The
demand matrices for trips that use the vehicle network are computed at every successive iteration by using the method
of successive averages over the vehicle flows. The solution algorithm flow chart is shown in the block diagram of
Figure 2.
Walk: M 3



fa( b) :
auto vehicles
Auto: M1
Transit: M 2
Network assignment
Auto assignment
Transit assignment
Walk assignment
fixed bus vehicles
Successive averaging
Find new vehicle volumes
pnm
Generalized costs Cij
pnm
ij
Travel demands T



Logsum computations
Matrix balancing
Demand computations
Trip distribution/mode choice
Figure 2. Block diagram of Equilibration Algorithm.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
The network block requires the solution of the variational inequality formulation subject to constraints (27) to (34) as
seen below. Since there are three distinct network modes M1, M2 , M3 , one obtains three corresponding variational
inequality problems which have equivalent optimization problems via a diagonalization. The network block is hence
given by:
   C
pnm *
( h , T )( hr
r
pn mM
ij
(26)
r Rijpnm
h
 Tijpnm , ( ij ), p, n, m
(27)
hrpnm  0, p, n, m  M , r  Rijpnm ,( ij ) W
(28)
pmn
r
s.t.
 hr* )  0, h (T ).
r
  h
pnm
 ar ,
r
f a1 
where
 a A
(29)
  h
pnm
 br ,
r
 b B
(30)
  h
pnm
 ar ,
r
 a A
(31)
pn mM1 r R1
f b2 
pn mM 2 r R2
f a3 
p mM3 r R3
Crpnm 
c
pnm
1
a ( f a ) ar ,
 p, n, m  M1, r  Rijpnm
(32)
aA
Crpnm 
c
pnm
( f a1( b) ) br ,
b
 p, n, m  M 2 , r  Rijpnm
(33)
bB
Crpnm 
c
pnm
a  ar ,
 p, n, m  M3 , r  Rijpnm
(34)
aA
The use of the nonlinear Jacobi method permits to solve the three distinct network modes in sequence. First, the auto
network problem (35) - (39) is solved as a fixed demand network equilibrium model for the network M1 .
  C
pnm *
( h , T )( hr
r
pn
s.t.
ij
mM1
 hr* )  0, h (T ).
(35)
r
h
pmn
r
 Tijpnm , ( ij ), p, n, m  M1
(36)
r
hrpnm  0, p, n, m  M1, r  Rijpnm
where
f a1 
  h
pnm
 ar ,
r
(37)
 a A
(38)
pn mM1 r R1
Crpnm 
c
pnm
1
a ( f a ) ar ,
 p, n, m  M1, r  Rijpnm
(39)
aA
Thus, the transit link costs can be determined to compute the transit flows (40) - (44) with link costs cbpnm ( f a1( b) ) to
compute the corresponding origin-destination costs.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
  C
pnm *
( h , T )( hr
r
pn
ij
mM2
(40)
r
h
pnm
r
s.t.
 hr* )  0, h (T ).
 Tijpnm , ( ij ), p, n, m  M 2
(41)
r
hrpnm  0, p, n, m  M2 , r
f b2 
where
  h
pnm
 br ,
r
(42)
 b B
(43)
 p, n, m  M 2 , r  Rijpnm
(44)
pn mM 2 r R2
Crpnm 
c
pnm
1
b ( f a ( b ) ) br ,
bB
Finally, the walk network model may be solved by an "all-or-nothing" assignment as the solution of (45) to (49).
  C
pnm *
( h , T )( hr
r
pn
s.t.
ij
mM3
 hr* )  0, h (T ).
(45)
r
h
pmn
r
 Tijpnm , ( ij ), p, n, m  M 3
(46)
r
hrpnm  0, p, n, m  M3 , r  Rijpnm
where
f a3 
  h
pnm
 ar ,
r
(47)
 a A
(48)
p mM3 r R3
Crpnm 
c
pnm
a  ar ,
 p, n, m  M3 , r  Rijpnm
(49)
aA
The demand block requires the computation of the multi-proportional trip distribution model described in the previous
sections, which is solved by an adaptation of the two-dimensional balancing method.
If  gp  1 ( g  G p ) and there are no constraints, (2), then (18) can be expressed as
Tijpnm  e 
pnm
 ipn  uij
pn
e
.
(50)
Using (1), (3) and (45), we obtain
e
 pn ipn
 Oipn /
  e 

g
m
pnm
ij
(51)
j
With (27) and (51), we have
Tijpnm  Oipn e
 uijpnm
/
 e
g
m'g
 uijpnm'
.
(52)
j
We call this case a “single constraint”. The logsum costs Lijpn are computed by straightforward matrix computations
according to the formulae stated in the previous section. The Equilibration Algorithm which employs the nonlinear
Jacobi method for the network blocks and the method of successive averages to dampen the demand is summarized
below.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
Equilibration Algorithm
Step 0
(Initialization): l  0 .
Choose on initial demand T 0 (which equals the sum of auto and taxi trips) and compute
Step 1
f 1, 0
(Computation of generalized costs): l  l 1 ,
1,l
1,l
Compute generalized costs C pnm,l ,( p, n, m) based on the current f a , ca ( f a ) and computed transit costs
cb ( fa1(,lb) ) .
Compute logsum matrices Lijpng and Lijpn ,( i , j ),( p , n , g ) .
Step 2
Step 3
(Combined trip distribution/mode choice model):
Carry out multi-proportional balancing (21)-(24) to obtain Tijpn , ( p , n ) and
Compute the O-D matrix of auto drivers T pnm,l ( p, n, m) (using the dual variables from the trip distribution
model).
(Vehicle network equilibrium assignment):
Compute (35)-(39) for
Step 4
f 1,l with Tijpnm( i , j )( p , n , m  M1 )
(Convergence test):
If f 1,l  f 1,l  (  0) , go to Step 6.
Step 5
(Method of Successive Averages):
Compute f 1,l 1  f 1,l ( x )  f 1,l (1  x ),( x  1 / ( l  1)) and return to Step 1.
Step 6
(Computation of final results): compute Tijpnm , ( i, j ), p , n , m
Compute final f 1, f 2 , f 3 .
Compute final Cijpnm , ( i, j ), p, n, m.
6. Computational results (STGO Model)
This model is implemented with EMME/2 and is applied in Santiago, Chile. The model is called STGO. And while
there are many details, such as parameters and equations (required for computation), it would serve no purpose to
discuss them here. However, we provide the following basic information to specify the STGO model. See Table 1 for
the STGO base network specification, Figure 3 for the network and Table 2 for the travel modes.
Table 1. STGO base network specifications
Supply Modes
12
Regular nodes
Travel Modes
11
Directional links
Centroids
Demands
264 Transit lines
zones
Parking zones 47
Transit line segments
Classes
13
Trip purposes
Table 2. Travel modes
Travel mode
Description
1 (wk)
Walk
2 (ad)
Auto-driver
3 (ap)
Auto-passenger
4 (tx)
Taxi
5 (tc)
Taxi-col
6 (bs)
Bus
Travel mode
7 (mt)
8 (bm)
9 (dm)
10 (pm)
11 (tm)
1091
5606
1273
45753
3
Description
Metro
Bus-metro
Auto-driver-metro
Auto-passenger-metro
Taxi-col-metro
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
Figure 3 Santiago Network
We note that different groups (nests) of travel modes are established according to trip purposes, with different
parameters. Here the EMME/2 implementation on a PC can effectively deal with the combined modes, such as park
and ride (see modes 9 and 10).
The implementation has the following features. Transit in-vehicle times depend on the vehicle travel time in the vehicle
network via transit functions. The generalized costs for transit related travel modes depend on the fares, mode specific
walk times, in-vehicle times, waiting times and boarding times.The generalized costs for vehicle related travel modes
depend on vehicle time and distances.The interactions between M1 , M 2 , M 3 are given in Figure 4 where the dash
lines imply no interactions implemented and the solid lines mean that there are interactions.In Step 1, the transit related
generalized costs are computed as done in the ESTRAUS model where five transit assignments are performed with
each for one transit mode. The corresponding computed components (such as in-vehicle times and fares) are used to
compute the generalized costs. In Step 3, the vehicle network equilibrium assignment (35)-(39) is performed as done in
the ESTRAUS model with a set of calibrated volume delayed functions. On a PC (PII 450, 96M), it took 20 min per
iteration. About ten iterations are needed to obtain very similar results to that of ESTRAUS, if the initial demand is
zero. If we want to perform a full run for a new scenario, we may only need three to five iterations.
Walk: M 3
fa( b) :
auto vehicles
Auto: M1
Transit: M 2
fixed bus vehicles
Figure 4. Interactions among the three assignment components.
The convergence of the Equilibration Algorithm for the STGO model is reported in Table 3. We see that after about ten
iterations, the total demand stabilizes. Figure 5 shows very similar demand totals by mode computed by both the
ESTRAUS and STGO models.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
Table 3. Convergence of the procedure (30 loops).
DEMAND
LINK VOLUME RELATIVE DIFFERENCE
LOOPS
demand
Average demand relative Total
difference (percentage)
max.
average
1
96.720695
0
3616
1
1
2
329579.5
94.80645
3962.34545
1
0.999583
3
321322
49.46157
1853.61218
55.236835
0.46832
…
…
…
…
…
…
9
294608.687
1.113139
874.864257
99.95684
0.222555
10
294363.687
0.832489
830.680847
89.338211
0.211154
11
294199.062
0.627432
782.620056
95.591217
0.198988
12
294104
0.53044
644.677001
77.937889
0.163957
…
…
…
…
…
…
30
293717.406
0.078193
333.711791
27.862077
0.084849
31
293736.562
0.05948
307.420928
28.45512
0.078164
demand
2000000
given by ESTRAUS
Computed by STGO
1500000
1000000
500000
total
pm
dm
tm
bm
mt
bs
tc
tx
ap
ad
wk
0
mode
Figure 5. Demand totals for each mode between STGO and ESTRAUS.
The advantages of using the variational inequality formulation are that it is more flexible and general by comparison
with the limited optimization formulation. Based on this formulation, we derived a Block Gauss-Seidel method. We
have not seen any optimization formulations developed so far which can result in the corresponding equations being
used in practice.
Acknowledgement: The authors would thank Prof. Maximo Bosch, Mr. Manuel Diaz, Mr. Christian Lopez and Mr. Fernando Bravo for all the
discussions during the project and Dr. Vladimir Livshits for a part of EMME/2 macro development.
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A Multi-Class Multi-Mode Variable Demand Equilibrium Model
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